Possibility (Sample Space) Diagrams

Possibility Diagrams

Possibility Diagrams

What is a possibility diagram?

• In probability, the sample space means all the possible outcomes

• In simple situations it can be given as a list

  • For flipping a coin, the sample space is: Heads, Tails

    • the letters H, T can be used

  • For rolling a six-sided dice, the sample space is:  1, 2, 3, 4, 5, 6 

• If there are two sets of outcomes, a grid can be used

  • These are called possibility diagrams (or sample space diagrams)

  • For example, roll two six-sided dice and add their scores

  • A list of all the possibilities would be very long

    • You might miss a possibility

    • It would be hard to spot any patterns in the sample space

• Combining more than two sets of outcomes must be done by listing the possibilities

  • For example, flipping three coins

    • The sample space is HHH, HHT, HTH, THH, HTT, THT, TTH, TTT (8 possible outcomes)

How do I use a possibility diagram to calculate probabilities?

• Probabilities can be found by counting the number of possibilities you want, then dividing by the total number of possibilities in the sample space

  • In the sample space 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, there are four prime numbers (2, 3, 5 and 7)

    • The probability of getting a prime number is  $\frac{4}{10}=\frac{2}{5}$

  • Using the possibility diagram above for rolling two dice, the probability of getting an eight is  $\frac{5}{36}$

    • There are 5 eights in the grid, out of the total 36 numbers

• Be careful, this counting method only works if all possibilities in the sample space are equally likely

  • For a fair six-sided dice: 1, 2, 3, 4, 5, 6 are all equally likely

  • For a fair (unbiased) coin: H, T are equally likely

  • Winning the lottery: Win, Lose are are not equally likely! 

    • You cannot count possibilities here to say the probability of winning the lottery is  $\frac{1}{2}$